Biomedical Engineering Reference
In-Depth Information
sensors. The solution only required relative
measures, so k i was normalized relative to the
amplitude of k 1 , the gain of the first channel.
Then, recalling that the machine performed 180°
rotation in 200 steps, we get
v 1 v 2 = P · F 1 P · F 2 ,
(9.2)
v 1 v 3 = P · F 1 P · F 3 .
(9.3)
Therefore,
200
k 1
k i =
n =0 | v 1 v 1 |
.
−1
(9.6)
F 1 F 2
F 1 F 3
v 1 v 2
v 1 v 3
200
n =0 | v i v i |
P T =
.
(9.4)
The factor k 1 / k i was used to correct the differ-
ence in b i between the channels, since q was com-
mon to all sensors and not dependent on sensor
parameters; hence,
The direction of polarization is given by
P = TAN −1 P x / P y . The solution in Eq. (9.4)
assumes a calibrated system for which F , q , and b
are known. The amplifier electronics and photo-
diode must be calibrated such that the three chan-
nels of the polarimeter are matched.
Considering differences between the photodi-
odes, our sensor model incorporated an unknown
scale factor for the conversion of light into volt-
age, k i for each of i diode channels; hence,
b i k 1
= v i k 1
k i b 1
k i v 1 .
(9.7)
The calibrated response c i of each sensor was
then given by
c i = k 1
b i k 1
k i v i
k i b 1
,
(9.8)
v i = b i + k i ( P · F i + q ).
(9.5)
without needing to determine the actual value
of any of the bias terms, yet ensuring that the
responses of all channels were matched to the
first channel.
The orientation of the polarization filters was
determined for each channel by computing the
phase of the sinusoidal response
We used the sinusoidal response of all three
polarimeter channels while they were directed
vertically upward and rotated about the vertical
axis (azimuth). The turntable moved discretely
at 200 steps per 180°. The three polarization fil-
ters were oriented at approximately 0°, 60°, and
120° to avoid parallel polarization-sensitivity
axes that would prevent the solution in Eq. (9.4)
from being well formed. Despite apparently
accurate manual alignment, the exact filter
polarization angles were not known well enough
to obtain a solution for P that would be useful
for aircraft navigation.
The relative gains between the channels were
established by removing the mean measure-
ment, v i , over a 180° rotation of each sensor,
which left a full-wavelength sinusoidal curve of
mean 0 for each of the i channels, with phase of
the sinusoid dependent on P and i th channel-
specific filter properties represented in F i . The
integral of the absolute values of each curve pro-
vided a distributed measure of the relative
amplitudes of the response from the two
200
( v i ) n v i
200
SIN n
200
F i =
n =0
(9.9)
200
( v i ) n v i
200
COS n
200
×
,
n =0
where 0. 5 F i is the orientation of the i ith channel
polarization filter.
So values of F i , k 1 / k i , and
were
calculated as above, and Eq. (9.4) was used to
solve for polarization direction. Figure 9.6 a
shows the signal from the sensors over a half-
rotation of the turntable.
The geometrical calibration step, in which F is
computed for each channel, is critical for
b i k k i b 1
 
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